Abstract

Ghost imaging could be used to make a quick identification of orthogonal objects by means of photocurrent correlation measurements. In this paper, we extend the method to identify nonorthogonal objects. In the method, an object is illuminated by one photon from an entangled pair, and the other one is diffracted into a particular direction by a pre-established multiple-exposure hologram in the idler arm. By the correlation measurements, the nonorthogonal object in the signal arm could be discriminated within a very short time. The constraints for the identification of nonorthogonal objects are presented, which show that the nonorthogonal objects can be discriminated when the overlapping portion between any two objects is less than half of all the objects in the set. The numerical simulations further verify the result.

© 2015 Chinese Laser Press

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References

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  1. B. Moghaddam and A. Pentland, “Probabilistic visual learning for object representation,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 696–710 (1997).
    [Crossref]
  2. C. Stauffer and W. E. L. Grimson, “Learning patterns of activity using real-time tracking,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 747–757 (2000).
    [Crossref]
  3. H. Vogt, “Efficient object identification with passive RFID tags,” in Pervasive Computing (Springer, 2002), pp. 98–113.
  4. Y. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13, 1016–1030 (2007).
    [Crossref]
  5. J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” in Quantum Information Processing (Springer, 2012), pp. 1–45.
  6. M. Malik, H. Shin, M. O’sullivan, P. Zerom, and R. W. Boyd, “Quantum ghost image identification with correlated photon pairs,” Phys. Rev. Lett. 104, 163602 (2010).
    [Crossref]
  7. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
    [Crossref]
  8. R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).
    [Crossref]
  9. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
    [Crossref]
  10. D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
    [Crossref]
  11. M. N. O’sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
    [Crossref]
  12. C. Wang, D. Zhang, Y. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
    [Crossref]
  13. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
    [Crossref]
  14. Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010).
    [Crossref]
  15. G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96, 063602 (2006).
    [Crossref]
  16. G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
    [Crossref]
  17. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
    [Crossref]
  18. Y. Bromberg and O. Katz, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
    [Crossref]

2010 (5)

M. Malik, H. Shin, M. O’sullivan, P. Zerom, and R. W. Boyd, “Quantum ghost image identification with correlated photon pairs,” Phys. Rev. Lett. 104, 163602 (2010).
[Crossref]

M. N. O’sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
[Crossref]

C. Wang, D. Zhang, Y. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[Crossref]

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010).
[Crossref]

2009 (1)

Y. Bromberg and O. Katz, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[Crossref]

2008 (1)

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[Crossref]

2007 (1)

Y. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13, 1016–1030 (2007).
[Crossref]

2006 (2)

G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96, 063602 (2006).
[Crossref]

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
[Crossref]

2005 (1)

D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
[Crossref]

2004 (1)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

2002 (1)

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).
[Crossref]

2000 (1)

C. Stauffer and W. E. L. Grimson, “Learning patterns of activity using real-time tracking,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 747–757 (2000).
[Crossref]

1997 (1)

B. Moghaddam and A. Pentland, “Probabilistic visual learning for object representation,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 696–710 (1997).
[Crossref]

1995 (1)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

Bai, Y.

C. Wang, D. Zhang, Y. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[Crossref]

Bennink, R. S.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).
[Crossref]

Bentley, S. J.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).
[Crossref]

Berardi, V.

G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96, 063602 (2006).
[Crossref]

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
[Crossref]

Boyd, R. W.

M. Malik, H. Shin, M. O’sullivan, P. Zerom, and R. W. Boyd, “Quantum ghost image identification with correlated photon pairs,” Phys. Rev. Lett. 104, 163602 (2010).
[Crossref]

M. N. O’sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
[Crossref]

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).
[Crossref]

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” in Quantum Information Processing (Springer, 2012), pp. 1–45.

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

Bromberg, Y.

Y. Bromberg and O. Katz, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[Crossref]

Cao, D.

D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
[Crossref]

Chan, K. W. C.

M. N. O’sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
[Crossref]

Chen, B.

C. Wang, D. Zhang, Y. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[Crossref]

Ferri, F.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

Gatti, A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

Grimson, W. E. L.

C. Stauffer and W. E. L. Grimson, “Learning patterns of activity using real-time tracking,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 747–757 (2000).
[Crossref]

Katz, O.

Y. Bromberg and O. Katz, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[Crossref]

Liu, J.

Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010).
[Crossref]

Lugiato, L. A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

Magatti, D.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

Malik, M.

M. Malik, H. Shin, M. O’sullivan, P. Zerom, and R. W. Boyd, “Quantum ghost image identification with correlated photon pairs,” Phys. Rev. Lett. 104, 163602 (2010).
[Crossref]

Moghaddam, B.

B. Moghaddam and A. Pentland, “Probabilistic visual learning for object representation,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 696–710 (1997).
[Crossref]

O’sullivan, M.

M. Malik, H. Shin, M. O’sullivan, P. Zerom, and R. W. Boyd, “Quantum ghost image identification with correlated photon pairs,” Phys. Rev. Lett. 104, 163602 (2010).
[Crossref]

O’sullivan, M. N.

M. N. O’sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
[Crossref]

Pentland, A.

B. Moghaddam and A. Pentland, “Probabilistic visual learning for object representation,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 696–710 (1997).
[Crossref]

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

Scarcelli, G.

G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96, 063602 (2006).
[Crossref]

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
[Crossref]

Sergienko, A. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

Shapiro, J. H.

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[Crossref]

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” in Quantum Information Processing (Springer, 2012), pp. 1–45.

Shih, Y.

Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010).
[Crossref]

Y. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13, 1016–1030 (2007).
[Crossref]

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
[Crossref]

G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96, 063602 (2006).
[Crossref]

Shih, Y. H.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

Shin, H.

M. Malik, H. Shin, M. O’sullivan, P. Zerom, and R. W. Boyd, “Quantum ghost image identification with correlated photon pairs,” Phys. Rev. Lett. 104, 163602 (2010).
[Crossref]

Simon, J.

Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010).
[Crossref]

Stauffer, C.

C. Stauffer and W. E. L. Grimson, “Learning patterns of activity using real-time tracking,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 747–757 (2000).
[Crossref]

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

Vogt, H.

H. Vogt, “Efficient object identification with passive RFID tags,” in Pervasive Computing (Springer, 2002), pp. 98–113.

Wang, C.

C. Wang, D. Zhang, Y. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[Crossref]

Wang, K.

D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
[Crossref]

Xiong, J.

D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
[Crossref]

Zerom, P.

M. Malik, H. Shin, M. O’sullivan, P. Zerom, and R. W. Boyd, “Quantum ghost image identification with correlated photon pairs,” Phys. Rev. Lett. 104, 163602 (2010).
[Crossref]

Zhang, D.

C. Wang, D. Zhang, Y. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[Crossref]

Zhou, Y.

Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010).
[Crossref]

Appl. Phys. Lett. (1)

G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88, 061106 (2006).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13, 1016–1030 (2007).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

B. Moghaddam and A. Pentland, “Probabilistic visual learning for object representation,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 696–710 (1997).
[Crossref]

C. Stauffer and W. E. L. Grimson, “Learning patterns of activity using real-time tracking,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 747–757 (2000).
[Crossref]

Phys. Rev. A (7)

D. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005).
[Crossref]

M. N. O’sullivan, K. W. C. Chan, and R. W. Boyd, “Comparison of the signal-to-noise characteristics of quantum versus thermal ghost imaging,” Phys. Rev. A 82, 053803 (2010).
[Crossref]

C. Wang, D. Zhang, Y. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010).
[Crossref]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[Crossref]

Y. Bromberg and O. Katz, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[Crossref]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995).
[Crossref]

Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010).
[Crossref]

Phys. Rev. Lett. (5)

G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. 96, 063602 (2006).
[Crossref]

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).
[Crossref]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

M. Malik, H. Shin, M. O’sullivan, P. Zerom, and R. W. Boyd, “Quantum ghost image identification with correlated photon pairs,” Phys. Rev. Lett. 104, 163602 (2010).
[Crossref]

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

Other (2)

H. Vogt, “Efficient object identification with passive RFID tags,” in Pervasive Computing (Springer, 2002), pp. 98–113.

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” in Quantum Information Processing (Springer, 2012), pp. 1–45.

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Figures (6)

Fig. 1.
Fig. 1. Schematic setup of quantum ghost image identification with correlated photon pairs.
Fig. 2.
Fig. 2. Procedure for establishing a two-times-exposure hologram. A light beam carrying the information of object a , denoted as object beam O 1 , first interferes with the reference beam R 1 on the hologram. Then, another light beam carrying the information of object b , denoted as object beam O 2 , interferes with another reference beam R 2 on the same hologram. So, the established two-times-exposure hologram contains the information of both object a and object b .
Fig. 3.
Fig. 3. Procedure of object discrimination using the multiple-exposure hologram. The information of object a and object b is already stored in the hologram. When the light beam carrying the information of object a , denoted as object beam O 1 , illuminates the hologram, the output beam is diffracted into the direction the same as the direction of the reference beam R 1 in the hologram establishing procedure. The same result is applicable to object b .
Fig. 4.
Fig. 4. Graph of the integral expressions. (a) Graphic form of t a ( x o ) t b ( x o ) . (b) Graphic form of t a 2 ( x o ) .
Fig. 5.
Fig. 5. Simulation results for the identification of two nonorthogonal objects. (a) Amplitude of object a and amplitude of object b . (b) Normalized coincidence results when object a or object b is placed in the signal arm.
Fig. 6.
Fig. 6. Simulation results for the identification of three nonorthogonal objects. (a) Amplitude of object a , amplitude of object b , and amplitude of object c . (b) Normalized coincidence results when object a , object b , or object c is placed in the signal arm.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

C = d x i d x s d x o d x h | f ( x i , x h ) f ( x s , x o ) T ( x o ) × H ( x h ) ψ ( x i , x s ) | 2 ,
I = | O ( x ) t 1 ( x ) + R 1 ( x ) | 2 + | O ( x ) t 2 ( x ) + R 2 ( x ) | 2 + | O ( x ) t n ( x ) + R n ( x ) | 2 ,
H ( x ) | O ( x ) t 1 ( x ) + R 1 ( x ) | 2 + | O ( x ) t 2 ( x ) + R 2 ( x ) | 2 + | O ( x ) t n ( x ) + R n ( x ) | 2 .
E out O t 1 ( x ) H ( x ) ( t 1 3 ( x ) | O | 2 + t 1 ( x ) | R | 2 ) O + t 1 2 ( x ) O 2 R 1 * + t 1 2 ( x ) | O | 2 R 1 + ( t 1 ( x ) t 2 2 ( x ) | O | 2 + t 1 ( x ) | R 2 | 2 ) O + t 1 ( x ) t 2 ( x ) O 2 R 2 * + t 1 ( x ) t 2 ( x ) | O | 2 R 2 + + ( t 1 ( x ) t n 2 ( x ) | O | 2 + t 1 ( x ) | R n | 2 ) O + t 1 ( x ) t n ( x ) O 2 R n * + t 1 ( x ) t n ( x ) | O | 2 R n .
H ( x ) t i ( x ) in R i derection .
C | d x o t a ( x o ) t b ( x o ) | 2 ,
C | d x o t a 2 ( x o ) | 2 .
4 | d x o t a ( x o ) t b ( x o ) | 2 | d x o t a 2 ( x o ) | 2 .
x 2 2 ( x 2 x 1 ) .

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